Classification of ancient ovals in higher dimensional mean curvature flow

TL;DR

The paper addresses the classification of ancient noncollapsed mean curvature flow solutions in $\mathbb{R}^{n+1}$, focusing on ancient ovals and the $k$-oval subclass defined by a cylindrical tangent flow $\mathbb{R}^k\times S^{n-k}$ and quadratic bending. Employing a spectral-parameter framework anchored in the Gaussian-weighted Hilbert space $\mathcal{H}$ and a pair of technical tools—a Jacobian-based transformation argument and a spectral map $\mathscr{E}$—the authors establish that any $k$-oval is, up to space-time rigid motion and parabolic dilations, an element of the Haslhofer–Du oval family $\mathcal{A}^{k,\circ}$. They prove a surjectivity result for the spectral map, enabling a prescribed spectral profile, and show the moduli space of ancient ovals is homeomorphic to the open $(k-1)$-simplex modulo the permutation group $\mathbf{S}_k$, providing a concrete parametrization of all such flows. While the broader classification was achieved in Bamler–Lai, the paper offers an alternative analytic route via spectral data and quadratic modes, reinforcing the structural picture and yielding a precise topological description of the ancient-oval moduli with potential applicability to related geometric-flow problems.

Abstract

We study compact non-selfsimilar ancient noncollapsed solutions to the mean curvature flow in $\mathbb{R}^{n+1}$, called ancient ovals. Our main result is the classification of $k$-ovals: any $k$-oval (characterized by having cylindrical blow down $\mathbb{R}^k\times S^{n-k}$ and the quadratic bending asymptotics) belongs, up to space-time rigid motions and parabolic dilations, to the family of ancient ovals constructed by Haslhofer and the second author. Assuming the nonexistence of exotic ovals (recently proved by Bamler-Lai), this yields a classification of all ancient ovals and identifies the moduli space, modulo symmetries, with an open $(k-1)$-simplex modulo the symmetry of simplex. Although these conclusions are contained in the recent breakthrough of Bamler-Lai classifying all ancient asymptotically cylindrical flows and resolving the mean convex neighborhood conjecture, we give an alternative argument for the independently obtained classification of $k$-ovals in arbitrary dimensions based on a different spectral parametrization.

Figures (3)

• Figure 1: $k=2$ case, left is mapped to right.
• Figure 2: $k=3$ case: left is mapped to the right.
• Figure 3: $k=4$ case: left is projected to the right.

Theorems & Definitions (46)

• Theorem 1.1: classification of $k$-ovals
• Conjecture 1.2: nonexistence of exotic ovals, now theorem by BL1BL2
• Theorem 1.3: classification of ancient ovals and moduli space
• Remark 1.4
• Definition 2.1: $k$-oval, c.f.DZ_spectral_quantization
• Definition 2.2: $\kappa$-quadraticity at $\tau_0$
• Definition 2.3: strong $\kappa$-quadraticity, c.f. CHH_translator
• Theorem 2.4: $\kappa$-quadraticity at one time implies strong $\kappa$-quadraticity c.f. CDZ_ovals
• Theorem 2.5: spectral uniqueness c.f. CDZ_ovals
• Proposition 3.1: orthogonality c.f. CDZ_ovals